Neutron Scattering - JUWEL - Forschungszentrum Jülich

HEiDi 7
The scattering factor F is a complex function describing the overlap of the scattering waves of
each atom i (n per unit cell). si(Q) describes the scattering strength of the i-th atom on its
position xi in dependence of the scattering vector Q, which depends on the character of cross
section as described below.
In this context one remark concerning statistics: For measurements of radiation the statistical
error � is the square root of the number of measured events, e.g. x-ray or neutron particles.
Thus, 100 events yield an error of 10% while 10,000 events yield an error of only 1%!
Mean Square Displacements (MSD): Thermal movement of atoms around their average
positions reduce the Bragg intensities during a diffraction experiment. The cause for this
effect is the reduced probability density and therefore reduced cross section probability at the
average positions. For higher temperatures (above a few Kelvin) the MSD Bi of the atoms
increase linearly to the temperature T, this means B ~ T. Near a temperature of 0 K the MSD
become constant with values larger than zero (zero point oscillation of the quantum
mechanical harmonic oscillator).
Thus, the true scattering capability si of the i-th atom in a structure has to be corrected by an
angle-dependent factor (the so called Debye-Waller factor):
si(Q) � si(Q) * exp(-Bi(sin �Q/�) 2 )
This Debye-Waller factor decreases with increasing temperatures and yields an attenuation of
the Bragg reflection intensities. At the same time this factor becomes significantly smaller
with larger sin���~|Q|. Therefore, especially reflections with large indices loose a lot of
intensity. The formula for anisotropic oscillations around their average positions looks like
this:
si(Q) � si(Q) * exp(-2� 2 (U i 11 h 2 a* 2 + U i 22 k 2 b* 2 + U i 33 l 2 c* 2 +
+ 2U i 13 hl a*c* + 2U i 12 hk a*b* + 2U i 23kl b*c*))
The transformation between B and Ueq (from the Uij calculated isotropic MSD for a sphere
with identical volume) yields B = 8� 2 Ueq.
For some structures the experimentally determined MSD are significantly larger than from the
harmonic calculations of the thermal movement of the atoms expected. Such deviations can
have different reasons: Static local deformations like point defects, mixed compounds,
anharmonic oscillations or double well potentials where two energetically equal atomic
positions are very near to each other and therefore distribute the same atom over the crystal
with a 50%/50% chance to one or the other position. In all those cases an additional
contribution to the pure Debye-Waller factor can be found which yields an increased MSD.
Therefore in the following text only the term MSD will be used to avoid misunderstandings.
3.2 Comparison of X-ray and Neutron Radiation
X-Ray Radiation interacts as electromagnetic radiation only with the electron density in a
crystal. This means the shell electrons of the atoms as well as the chemical binding. The
scattering capability s (atomic form factor f(sin���)) of an atom depends on the number Z of
its shell electrons (f(sin(�=0)/�) =Z). To be exact, f(sin(�)/�) is the Fourier transform of the
radial electron density distribution ne(r): f(sin(�)/�)=s � � 0 4� 2 ne(r) sin(μr)/μr dr with
μ=4�sin(�)/���Heavy atoms with many electrons contribute much stronger to reflection